Quiz — Kounouz Ennajeh 4math Partie2.pdf

Question 1 sur 10 20:00

[{"id":15159,"question":"Quelle est la dérivée de la fonction f(x) = 3x² + 2x - 5 ?","option_a":"6x + 2","option_b":"3x² + 2","option_c":"6x - 5","option_d":"6x + 2x - 5","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée d'une fonction polynôme se calcule terme par terme : la dérivée de 3x² est 6x, celle de 2x est 2, et celle de -5 est 0. D'où f'(x) = 6x + 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"6x + 2\", \"b\": \"3x² + 2\", \"c\": \"6x - 5\", \"d\": \"6x + 2x - 5\"}}","_debug_options_count":4},{"id":15160,"question":"L'équation différentielle y' = 2y admet pour solution générale y = Ce^(2x), où C est une constante réelle.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La solution générale de y' = ky est y = Ce^(kx). Ici, k = 2, donc la solution est bien y = Ce^(2x).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":15161,"question":"Quel est le module d'un nombre complexe z = 3 + 4i ?","option_a":"5","option_b":"7","option_c":"√7","option_d":"√(3² + 4²)","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le module d'un nombre complexe z = a + bi est donné par |z| = √(a² + b²). Ici, |z| = √(3² + 4²) = 5.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"5\", \"b\": \"7\", \"c\": \"√7\", \"d\": \"√(3² + 4²)\"}}","_debug_options_count":4},{"id":15162,"question":"La probabilité d'obtenir exactement 2 fois pile en 3 lancers d'une pièce non truquée est :","option_a":"1\/4","option_b":"3\/8","option_c":"1\/2","option_d":"1\/8","option_e":"","option_f":"","bonne_reponse":"b","explication":"C'est une loi binomiale de paramètres n=3 et p=1\/2. P(X=2) = C(3,2) * (1\/2)^2 * (1\/2)^1 = 3\/8.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"1\/4\", \"b\": \"3\/8\", \"c\": \"1\/2\", \"d\": \"1\/8\"}}","_debug_options_count":4},{"id":15163,"question":"Une matrice carrée A est inversible si et seulement si son déterminant est nul.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Une matrice est inversible si et seulement si son déterminant est différent de zéro. Si le déterminant est nul, la matrice n'est pas inversible.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":15164,"question":"Quelle est la solution générale de l'équation différentielle y'' + y = 0 ?","option_a":"y = Ce^x","option_b":"y = C1e^x + C2e^(-x)","option_c":"y = C1cos(x) + C2sin(x)","option_d":"y = Cx + D","option_e":"","option_f":"","bonne_reponse":"c","explication":"L'équation caractéristique associée est r² + 1 = 0, dont les racines sont r = i et r = -i. La solution générale est donc y = C1cos(x) + C2sin(x).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"y = Ce^x\", \"b\": \"y = C1e^x + C2e^(-x)\", \"c\": \"y = C1cos(x) + C2si","_debug_options_count":4},{"id":15165,"question":"Si A = [[1, 2], [3, 4]], alors A² = ?","option_a":"[[7, 10], [15, 22]]","option_b":"[[1, 4], [9, 16]]","option_c":"[[3, 6], [9, 12]]","option_d":"[[1, 2], [3, 4]]","option_e":"","option_f":"","bonne_reponse":"a","explication":"A² = A × A = [[1*1 + 2*3, 1*2 + 2*4], [3*1 + 4*3, 3*2 + 4*4]] = [[7, 10], [15, 22]].","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"[[7, 10], [15, 22]]\", \"b\": \"[[1, 4], [9, 16]]\", \"c\": \"[[3, 6], [9","_debug_options_count":4},{"id":15166,"question":"La variance d'une variable aléatoire X est toujours positive ou nulle.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La variance mesure la dispersion des valeurs autour de la moyenne. Par définition, elle est toujours positive ou nulle (égale à zéro si toutes les valeurs sont identiques).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":15167,"question":"Quelle est la forme trigonométrique du nombre complexe z = -1 + i√3 ?","option_a":"2(cos(π\/3) + i sin(π\/3))","option_b":"2(cos(2π\/3) + i sin(2π\/3))","option_c":"√3(cos(π\/3) + i sin(π\/3))","option_d":"2(cos(π\/6) + i sin(π\/6))","option_e":"","option_f":"","bonne_reponse":"b","explication":"Le module de z est √((-1)² + (√3)²) = 2. L'argument θ vérifie cos(θ) = -1\/2 et sin(θ) = √3\/2, donc θ = 2π\/3. D'où z = 2(cos(2π\/3) + i sin(2π\/3)).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"2(cos(π\/3) + i sin(π\/3))\", \"b\": \"2(cos(2π\/3) + i sin(2π\/3))\",","_debug_options_count":4},{"id":15168,"question":"La somme des racines d'un polynôme du second degré ax² + bx + c est égale à :","option_a":"-b\/a","option_b":"b\/a","option_c":"-c\/a","option_d":"c\/a","option_e":"","option_f":"","bonne_reponse":"a","explication":"Pour un polynôme ax² + bx + c, la somme des racines est donnée par -b\/a (relation de Viète).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"-b\/a\", \"b\": \"b\/a\", \"c\": \"-c\/a\", \"d\": \"c\/a\"}}","_debug_options_count":4}]